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The Stacks project

Lemma 104.3.2. With assumptions and notation as in Cohomology of Stacks, Lemma 103.15.1. We have

g^{-1} \circ Rf_* = Rf'_* \circ (g')^{-1} \quad \text{and}\quad L(g')_! \circ (f')^{-1} = f^{-1} \circ Lg_!

on unbounded derived categories (both for the case of modules and for the case of abelian sheaves).

Proof. Let \tau = {\acute{e}tale} (resp. \tau = fppf). Let \mathcal{F} be an abelian sheaf on \mathcal{X}_\tau . By Cohomology of Stacks, Lemma 103.15.3 the canonical (base change) map

g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F}

is an isomorphism. The rest of the proof is formal. Since cohomology of abelian groups and sheaves of modules agree we also conclude that g^{-1} Rf_*\mathcal{F} = Rf'_* (g')^{-1}\mathcal{F} when \mathcal{F} is a sheaf of modules on \mathcal{X}_\tau .

Next we show that for \mathcal{G} (either sheaf of modules or abelian groups) on \mathcal{Y}_{lisse,{\acute{e}tale}} (resp. \mathcal{Y}_{flat,fppf}) the canonical map

L(g')_!(f')^{-1}\mathcal{G} \to f^{-1}Lg_!\mathcal{G}

is an isomorphism. To see this it is enough to prove for any injective sheaf \mathcal{I} on \mathcal{X}_\tau the induced map

\mathop{\mathrm{Hom}}\nolimits (L(g')_!(f')^{-1}\mathcal{G}, \mathcal{I}[n]) \leftarrow \mathop{\mathrm{Hom}}\nolimits (f^{-1}Lg_!\mathcal{G}, \mathcal{I}[n])

is an isomorphism for all n \in \mathbf{Z}. (Hom's taken in suitable derived categories.) By the adjointness of f^{-1} and Rf_*, the adjointness of Lg_! and g^{-1}, and their “primed” versions this follows from the isomorphism g^{-1} Rf_*\mathcal{I} \to Rf'_* (g')^{-1}\mathcal{I} proved above.

In the case of a bounded complex \mathcal{G}^\bullet (of modules or abelian groups) on \mathcal{Y}_{lisse,{\acute{e}tale}} (resp. \mathcal{Y}_{fppf}) the canonical map

104.3.2.1
\begin{equation} \label{stacks-perfect-equation-to-show} L(g')_!(f')^{-1}\mathcal{G}^\bullet \to f^{-1}Lg_!\mathcal{G}^\bullet \end{equation}

is an isomorphism as follows from the case of a sheaf by the usual arguments involving truncations and the fact that the functors L(g')_!(f')^{-1} and f^{-1}Lg_! are exact functors of triangulated categories.

Suppose that \mathcal{G}^\bullet is a bounded above complex (of modules or abelian groups) on \mathcal{Y}_{lisse,{\acute{e}tale}} (resp. \mathcal{Y}_{fppf}). The canonical map (104.3.2.1) is an isomorphism because we can use the stupid truncations \sigma _{\geq -n} (see Homology, Section 12.15) to write \mathcal{G}^\bullet as a colimit \mathcal{G}^\bullet = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ n^\bullet of bounded complexes. This gives a distinguished triangle

\bigoplus \nolimits _{n \geq 1} \mathcal{G}_ n^\bullet \to \bigoplus \nolimits _{n \geq 1} \mathcal{G}_ n^\bullet \to \mathcal{G}^\bullet \to \ldots

and each of the functors L(g')_!, (f')^{-1}, f^{-1}, Lg_! commutes with direct sums (of complexes).

If \mathcal{G}^\bullet is an arbitrary complex (of modules or abelian groups) on \mathcal{Y}_{lisse,{\acute{e}tale}} (resp. \mathcal{Y}_{fppf}) then we use the canonical truncations \tau _{\leq n} (see Homology, Section 12.15) to write \mathcal{G}^\bullet as a colimit of bounded above complexes and we repeat the argument of the paragraph above.

Finally, by the adjointness of f^{-1} and Rf_*, the adjointness of Lg_! and g^{-1}, and their “primed” versions we conclude that the first identity of the lemma follows from the second in full generality. \square


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